We say that a statement, or set of statements is logically consistent
when
it involves no logical contradiction. A logical contradiction
is the conjunction of a statement S and its denial not-S.
In logic, it is a fundamental law- the law of non contradiction- that a
statement and its denial cannot both be true at the same time. Here
are some simple examples of contradictions.

1. I love you and I don't love you.
2. Butch is married to Barb but Barb is not married to Butch.
3. I know I promised to show up today, but I don't see why I
should come if I don't feel like it.
4. The restaurant opens at five o'clock and it begins serving
between four and nine.
5. John Lasagna will be a little late for the party. He
died yesterday.

These all seem to be contradictions because they seem either explicitly
to state or logically imply a certain statement and its denial. (1) is
an explicit contradiction. You can't love someone and not love someone
at the same time. (2) is an implicit contradiction. It depends
on the unstated but well known principle: if x is married to y, then
y is married to x. (3) is also an implicit contradiction. It
depends on the unstated meaning of promising, namely, that whenever you
promise to do something you thereby acquire a moral obligation to do it.

Very often contradictions are only apparent. For example someone
in a love-hate relationship might utter something like (1), meaning "I
love you in some ways, but I hate you in others". This, of course,
is not a contradiction at all. (4) also can look like a contradiction, but
this may just be the result of unclarity. Perhaps the restaurant
opens at 5:00 in the morning. (5) is not literally a contradiction, since
a dead person could show up at a party. We call it a contradiction
just because the statement "John will be a little late for the party,"
strongly suggests that John will be alive when he shows up.

When we tell people that they aren't making any sense, it is often because
we think that they are saying something contradictory. In a Dilbert
cartoon one of Dilbert's office mates is complaining that she hasn't been
trained how to use the new computer. The conversation proceeds as follows:

Dilbert: Why don't you just read the manual?

Office mate: Right. Who has time to do that?

Dilbert: You mean you have time to go to a training session, but
you don't have time to read the manual? That doesn't add up.

Here Dilbert's point is obviously that she is contradicting herself :
She is saying that she has time to learn and she doesn't. Of course
she might not really be contradicting herself at all. It may be
that she finds computer manuals very hard to understand, so that the time
it takes to be trained really is far less than the time it would take to
learn from the manual.

This example shows that while it is very important to be logically consistent,
it is also important to permit people to be so. When people speak
in a way that seems logically contradictory it is often just because
they are not speaking completely or clearly. So the point of exposing
apparent contradictions is not, ultimately, to criticize peoples views
as nonsensical, but rather to make them be clearer about what they are
saying.

Even when someone really is contradicting himself we often tend to go
too far in the criticism. For example, if I were to tell you that
I do not like to eat fish, and you knew that I often ate and enjoyed tuna
salad sandwiches, you would be justified in pointing out that I was involved
in a logical contradiction and you might reason as follows:

"I know you like tuna sandwiches, and tuna is fish, so you are saying that
you both do and don't like fish. That makes no sense."

It is important to observe, however, that you would not have been
justified in responding as follows:

"I know you like tuna sandwiches and since tuna is a fish you do like at
least one fish."

The difference between these two responses is subtle but important.
In the first case you have informed me, correctly, that I can not both
like tuna and not like tuna at the same time, but you allowed me to make
up my mind about which statement I am going to revise. In the second
case you assumed that my commitment to liking tuna is stronger than my
commitment to not liking fish, and instructed me to revise my view that
I don't like fish. But suppose that the information that tuna is
a fish made me sick to my stomach- perhaps I had never fully appreciated
this fact- I may, as a consequence, never touch tuna again. In that
case, I would have revised the statement that I like tuna and thereby preserved
the generalization that I don't like fish . You might consider this to
be an absurd outcome, but it's not actually logically contradictory, because
its not a contradiction to like tuna at one time and not like it at another.

The point here is that exposing a logical contradiction is just the
beginning of a useful criticism. If the contradiction we point out is real,
then we have essentially challenged the speaker to revise his or her views
in one of several possible ways. A full, fair, critique, must take account
of the various possible ways that this can be done.

Consider another example: Suppose I believe that atheists are bad people,
and that all my friends are good people. But Mr. Pheeper, my
long-time friend, decides that he is an atheist. I am now faced with
accepting the following list of statements:

(a) Mr. Pheeper is my friend
(b) All my friends are good people.
(c) Mr. Pheeper is an atheist.
(d) All atheists are bad people.

These four statements are logically contradictory, because they jointly
imply that Mr. Pheeper is both a good person and a bad person. Logic requires
some sort of revision to my set of beliefs, but logic does not demand one
particular revision. I could (a) decide that Mr. Pheeper is no longer my
friend, (b) decide that atheists aren't necessarily bad people, (c) decide
that not all my friends are good people, or (d) decide Mr. Pheeper
is not an atheist even though he says that he is. The point is that these
are all possible solutions, each of which must be examined on their own
merits.

Everyone understands at some level that contradictions must be avoided
because they don't make any sense; so it is rare for people who understand
what they are saying to contradict themselves explicitly. Usually contradictions
are implicit in (i.e., logically implied by) what someone says. This means
that there is usually a generalization, or principle, that the person is
committed to which implies a statement that is inconsistent with (a) particular
facts (like the tuna example) or (b) statements implied by other principles
that the person also accepts (like the atheist example.)

Identifying Contradictions

Most people find it difficult to identify contradictions in an explicit
way, but it is important to learn to do so. Here is an explicit
definition of a contradiction together with the proper method for identifying
one.

Contradiction

Def.: To be logically committed to the assertion of some statement,
S, and its denial, not-S, at the same time.
ID.: Identify the statement that being both asserted and denied.

To see how to identify contradictions properly, consider the following
conversational example.

Mrs. Beeble: You have been absent from class 11 times this month.
You fail. Goodbye.

Butch: What? That's impossible! The class only
meets twice a week.

Mrs. Beeble: True, but you have missed it 11 times nevertheless.

Butch: That doesn't make any sense. You can't be absent from
a class on days that it doesn't meet.

Mrs. Beeble: You're saying you weren't absent on those days?

Butch: That's right..

Mrs. Beeble: So you were here?

Butch: No!

Mrs. Beeble: You are wasting my time. If you weren't here,
then you were absent. Goodbye.

Notice here that Butch, while perfectly aware that Mrs. Beeble is talking
nonsense, has not successfully identified the contradiction. The
closest he comes is to say "You can't be absent from a class on days
that it doesn't meet." But this statement can have several different
meanings. Butch is trying to say that it is logically contradictory
to mark someone absent from a class that doesn't meet. But all he
is succeeded in communicating is that it is somehow an unfair or unreasonable
requirement, which is open to interpretation. Here then is a better
way for Butch to proceed:

Mrs. Beeble: You have been absent from class 11 times this month.
You fail. Goodbye.

Butch: What? That's impossible! The class only
meets twice a week.

Mrs. Beeble: True, but you have missed it 11 times nevertheless.

Butch: So you are saying that I was absent on days that the class
didn't meet.

Mrs. Beeble: That's right.

Butch: Well, then you have committed a logical error. "X was
absent from event Y," logically implies that event Y occurred.
So you are saying that on the days that I was absent, the class both met
and did not meet. That is nonsense.

Mrs. Beeble: Sit down and shut up you little bastard.

Actually, the truth is that proper logical analysis does not compel agreement
anymore than Butch's first approach. Many people find it tiresome
and offensive, but logic doesn't concern itself with things like that.

Consistency and Deductive Implication

Logical consistency is essential to good reasoning, but it is by no
means sufficient. Completely invalid reasoning will be logically
consistent if the statements simply have nothing to do with each other.
For example the following argument

1. Some dogs have fleas.2. Therefore I want a Reese's Peanut Butter Cup.

is logically consistent. So, like deductive implication, the fact
that an argument is logically consistent isn't always interesting.

The concept of contradiction does, however, give us an interesting way
of defining the idea of deductive implication. We know that reasoning
is deductively valid whenever it is impossible for the premise(s) to be
true and the conclusion false. Another way of saying this is that
a logical contradiction arises when we assume that the premises
are true but the conclusion is false. For example the following reasoning
is valid

(1) All philosophers are transvestites.
(2) Melvin is a philosopher.(3) Therefore Melvin is a transvestite.

because if we assume that Melvin is not a transvestite this contradicts
(1) and (2) which jointly imply that he is a transvestite.

On the other hand

(1) Everyone over the age of 30 is a liar.
(2) Mr. Pheepher is a liar.(3) Therefore Mr. Pheepher is over the age of 30.

strikes many people as deductively valid. It is not,
however, because if we assume that Mr. Pheepher is not over the age of
30, no contradiction arises. This sort of thing can be difficult
to see intuitively, so in order to show that no contradiction arises we
can offer a counterexample. A counterexample is a state of
affairs in which the premises are true and the conclusion is false.
A counterexample to the above situation would be a world in which everyone
lies. In this case everyone over the age of 30 remains a liar, but so is
everyone 30 and under, so there is no contradiction in assuming that Mr.
Pheepher is 30 or under.